TY - JOUR
T1 - Optimal separable bases and series expansions
AU - Poirier, Bill
PY - 1997
Y1 - 1997
N2 - A method is proposed for the efficient calculation of the Green’s functions and eigenstates for quantum systems of two or more dimensions. For a given Hamiltonian, the best possible separable approximation is obtained from the set of all Hilbert-space operators. It is shown that this determination itself, as well as the solution of the resultant approximation, is a problem of reduced dimensionality. Moreover, the approximate eigenstates constitute the optimal separable basis, in the sense of self-consistent field theory. The full solution is obtained from the approximation via iterative expansion. In the time-independent perturbation expansion for instance, all of the first-order energy corrections are zero. In the Green’s function case, we have a distorted-wave Born series with optimized convergence properties. This series may converge even when the usual Born series diverges. Analytical results are presented for an application of the method to the two-dimensional shifted harmonic-oscillator system, in the course of which the quantum [Formula Presented] potential problem is solved exactly. The universal presence of bound states in the latter is shown to imply long-lived resonances in the former. In a comparison with other theoretical methods, we find that the reaction path Hamiltonian fails to predict such resonances.
AB - A method is proposed for the efficient calculation of the Green’s functions and eigenstates for quantum systems of two or more dimensions. For a given Hamiltonian, the best possible separable approximation is obtained from the set of all Hilbert-space operators. It is shown that this determination itself, as well as the solution of the resultant approximation, is a problem of reduced dimensionality. Moreover, the approximate eigenstates constitute the optimal separable basis, in the sense of self-consistent field theory. The full solution is obtained from the approximation via iterative expansion. In the time-independent perturbation expansion for instance, all of the first-order energy corrections are zero. In the Green’s function case, we have a distorted-wave Born series with optimized convergence properties. This series may converge even when the usual Born series diverges. Analytical results are presented for an application of the method to the two-dimensional shifted harmonic-oscillator system, in the course of which the quantum [Formula Presented] potential problem is solved exactly. The universal presence of bound states in the latter is shown to imply long-lived resonances in the former. In a comparison with other theoretical methods, we find that the reaction path Hamiltonian fails to predict such resonances.
UR - http://www.scopus.com/inward/record.url?scp=0001618414&partnerID=8YFLogxK
U2 - 10.1103/PhysRevA.56.120
DO - 10.1103/PhysRevA.56.120
M3 - Article
AN - SCOPUS:0001618414
SN - 1050-2947
VL - 56
SP - 120
EP - 130
JO - Physical Review A - Atomic, Molecular, and Optical Physics
JF - Physical Review A - Atomic, Molecular, and Optical Physics
IS - 1
ER -